alex your Spanish sentence should be in that case : Nosotros, los reyes, escribimos las cartas. In your way, the subject is NOSOTROS and escribimos becomes ok. So, the pronoun and the comas, are important.

The "Nosotros" is not needed, if you are a king you can say "Los reyes escribimos cartas". Trust me, I'm Spanish... We almost never say the pronoun... it is redoundant if you say "escribimos", we know "escribimos" goes with "nosotros"...